This paper is concerned
with upper semicontinuous decompositions of the 3-sphere which have the property
that the closure of the sum of the nondegenerate elements projects onto a set
which is 0-dimensional in the decomposition space. It is shown that such a
decomposition is definable by cubes with handles if it is point-like. This fact
is then used to obtain some properties of point-like decompositions of the
3-sphere which imply that the decomposition space is a topological 3-sphere.
It is also shown that decompositions of the 3-sphere which are definable
by cubes with one hole must be pointlike if the decomposition space is a
3-sphere.
